Segmentation of heterogeneous document images : an ... - Tel

weights that are assigned to edge and node feature functions, respectively.
A node feature function f n judges how likely a label y s is assigned to site
s given the observations x s . An edge feature function f e indicates how likely
two adjacent sites have particular label configuration given the observations at
both sites. Feature functions and their weights can take any arbitrary real value.
With a slight abuse of notation, it is usual to write this probability in a compact
form as below:
( F
)
∑
p s (y s |x s ) ∝ exp λ k f k (y i,i∈N(s) , y s , x i,i∈N(s) , x s )
k=1
tel-00912566, version 1 - 2 Dec 2013
where F = F e + F n is the total number of feature functions. According
to the Hammersley-Clifford theorem (1971), the conditional probability of a set
of sites given a set of observations is proportional to the product of potential
functions on cliques of the graph. If we take cliques to be pair of adjacent sites
then:
(
p(y|x) = 1 ∏ F
)
∑
exp λ k f k (y i,i∈N(s) , y s , x s , x i,i∈N(s) , x s )
Z
s∈S
k=1
The scalar Z is the normalization factor, or partition function, to ensure
that the probability is valid. It is defined as the sum of exponential terms for all
possible label configurations. Computing such partition function is intractable
for most applications due to computational costs. As a consequence, by assuming
conditional independence of y’s given x’s, the conditional probability of the
CRF model for the whole image can be defined:
p(y|x) = ∏ ( F
)
1 ∑
exp λ k f k (y i,i∈N(s) , y s , x s , x i,i∈N(s) , x s )
Z s
s∈S
k=1
Then Z for each site becomes:
Z s =
∑
(y 1,y 2)∈Y 2 exp
(
∑ F
)
λ i f i (y 1 , y 2 , x s , x N(s) )
i=1
where Y = {text, non-text}. Notice that the summation in partition function
is still composed of 512 terms for a 3 × 3 sites.
4.1.1 The three basic problems for CRFs
Given the CRF model of the previous section, there are three basic problems
of interest that must be solved for the model to be useful in real-world applications.
These problems are the following:
• Problem 1: Given all the observations x, the label configuration y and a
model ψ = (f, λ), how do we efficiently compute p(y|x, ψ), the conditional
probability of label configuration given the model? (Marginal inference)
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